math/greatest_common_divisor: add support for negative numbers (#2628)

* add type hints to math/gcd

* add doctest

* math/gcd - run black formatter

* math/gcd: remove manual doctest

* add correction to gcd of negative numbers

* add more doctest in iterative gcd

maths/greatest_common_divisor.py

@@ -2,10 +2,12 @@
 Greatest Common Divisor.
 
 Wikipedia reference: https://en.wikipedia.org/wiki/Greatest_common_divisor
+
+gcd(a, b) = gcd(a, -b) = gcd(-a, b) = gcd(-a, -b) by definition of divisibility
 """
 
 
-def greatest_common_divisor(a, b):
+def greatest_common_divisor(a: int, b: int) -> int:
     """
     Calculate Greatest Common Divisor (GCD).
     >>> greatest_common_divisor(24, 40)
@@ -20,31 +22,44 @@ def greatest_common_divisor(a, b):
     1
     >>> greatest_common_divisor(16, 4)
     4
+    >>> greatest_common_divisor(-3, 9)
+    3
+    >>> greatest_common_divisor(9, -3)
+    3
+    >>> greatest_common_divisor(3, -9)
+    3
+    >>> greatest_common_divisor(-3, -9)
+    3
     """
-    return b if a == 0 else greatest_common_divisor(b % a, a)
+    return abs(b) if a == 0 else greatest_common_divisor(b % a, a)
 
 
-"""
+def gcd_by_iterative(x: int, y: int) -> int:
-Below method is more memory efficient because it does not use the stack (chunk of
-memory).  While above method is good, uses more memory for huge numbers because of the
-recursive calls required to calculate the greatest common divisor.
-"""
-
-
-def gcd_by_iterative(x, y):
     """
+    Below method is more memory efficient because it does not create additional
+    stack frames for recursive functions calls (as done in the above method).
     >>> gcd_by_iterative(24, 40)
     8
     >>> greatest_common_divisor(24, 40) == gcd_by_iterative(24, 40)
     True
+    >>> gcd_by_iterative(-3, -9)
+    3
+    >>> gcd_by_iterative(3, -9)
+    3
+    >>> gcd_by_iterative(1, -800)
+    1
+    >>> gcd_by_iterative(11, 37)
+    1
     """
     while y:  # --> when y=0 then loop will terminate and return x as final GCD.
         x, y = y, x % y
-    return x
+    return abs(x)
 
 
 def main():
-    """Call Greatest Common Divisor function."""
+    """
+    Call Greatest Common Divisor function.
+    """
     try:
         nums = input("Enter two integers separated by comma (,): ").split(",")
         num_1 = int(nums[0])